It's a line that echoes across the campaign trail this season. Mr. Pryor, son of former Arkansas Sen. David Pryor (D), is one of a number of sons and daughters attempting to follow parents into political office  often, into the very same seats.

To name just a few: In Texas, the son of retiring House majority leader Dick Armey (R) is running in his father's district. In New York and Pennsylvania, the sons of former Govs. Mario Cuomo (D) and Bob Casey (D) are competing for governorships. Even Al Gore's oldest daughter, Karenna Gore Schiff, told CNN last week that she "wouldn't rule out" a run for office.

In some ways, political dynasties are the closest this country gets to royalty  and at a time when the current president is the son of a former president, the trend seems more prominent than ever.

These candidates' advantages are obvious: a brand name, and often, an inherited political organization and a network of contributors.

But there can be disadvantages, too.

Candidates sometimes seem diminished in the shadows of famous parents  as George W. Bush discovered early in 2000, when his father campaigned for him in New Hampshire and called him "this boy, this son of ours."

In last month's Texas primary, Scott Armey didn't win enough votes to avoid today's runoff election. And Rep. Ken Bentsen (D), nephew of former Sen. Lloyd Bentsen, lost his bid for the Senate.

Most of the time, being the child of a popular politician is "worth one step up the ladder," says Stephen Hess, author of "America's Political Dynasties" and a senior fellow at the Brookings Institution. "Then you've got to make it on your own."

Political dynasties are nothing new. Ever since the days of John Adams and John Quincy Adams, most states have had at least one family that's sent generations of politicians to Congress  the Frelinghuysens of New Jersey, for example, and the Longs of Louisiana.

Some families spill into multiple states, as the Kennedys have done from Massachusetts to Rhode Island to Maryland (where the latest contender, Democrat Kathleen Kennedy Townsend, is now running for governor).

Though there are no reliable statistics on dynasties, Mr. Hess says the number seems to have remained fairly constant over time. On the one hand, he points out, the trend toward smaller families has probably lowered the odds of getting "one [child] who will go into Daddy's business."

Yet other factors  such as the number of women going into politics  widen the familial pool. Not only are daughters of politicians running for office, but more and more wives are, as well  such as Elizabeth Dole (R) in North Carolina, and Sen. Jean Carnahan (D) in Missouri.

And there are new black and Hispanic dynasties, such as the Fords (D) of Tennessee and the Gonzalezes (D) of Texas.

Moreover, analysts say, as campaigning relies more on soundbites and ads, the value of famous names may be growing.

"Sometimes, it even helps people who aren't related but who have the same last name," points out Donald Ritchie, associate Senate historian.

Of course, this can cut both ways. In Illinois, state Attorney General and gubernatorial candidate Jim Ryan (R) may have to work hard to convince voters that he is not related to current Gov. George Ryan (R), whose tenure has been marred by scandal.

And Illinois Rep. Jesse L. Jackson, Jr. (D) (another son of a politician) went to court earlier this year to get a challenger named "Jesse L. Jackson" off the ballot.

While dynasties may seem a somewhat undemocratic phenomenon, analysts say it's not surprising that so many children of politicians tend toward public lives of their own, having grown up in environment heavy on political chatter.

Moreover, Hess adds, this background often makes dynasts "pretty good public servants, by and large  probably better, on average, than non-dynasts."

Children of politicians are often acutely aware of parents' legacies, and try to live up to them  or atone for them, says Mr. Ritchie. Henry Cabot Lodge Jr., whose grandfather was largely responsible for killing the League of Nations, went on to become ambassador to the UN.

Former Senator Pryor says that when his three sons were growing up, he took them to watermelon suppers and pie suppers around Arkansas.

"We would haul them around unmercifully to all these events," he recalls, laughing. "We thought they would grow up to be abnormal, but they're three fairly normal human beings. But on Mark, I think it just kind of took."

Mr. Pryor says he's well aware of the potential pitfalls for sons of politicians, particularly if parents seem overly involved.

He and his son debated, for example, how prominently he should be featured in the recent televised ad. (He's in it, but he doesn't speak.)

And his role in the campaign will be limited, he says, to "head of the sign crew."

Nor does he believe that lineage would give his son an easier time once in office.

"As one of the Udalls put it," Pryor says, referring to Rep. Tom Udall (D) of New Mexico (son of former Rep. Stewart Udall) and his cousin, Rep. Mark Udall (D) of Colorado (son of former Rep. Morris Udall), " 'You usually get half your dad's friends, and all of his enemies.'"